Lectures on Mathematics (New York, 1911), p. 8.
[1927]. Every one knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.... A curve is the totality of points, whose co-ordinates are functions of a parameter which may be differentiated as often as may be required.—Klein, F.
Elementar Mathematik vom höheren Standpunkte aus. (Leipzig. 1909) Vol. 2, p. 354.
[1928]. Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along telegraph wires, and the conduction of heat by the earth’s crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance.—Thomson and Tait.
Elements of Natural Philosophy, chap. 1.
[1929]. The principal advantage arising from the use of hyperbolic functions is that they bring to light some curious analogies between the integrals of certain irrational functions.—Byerly, W. E.
Integral Calculus (Boston, 1890), p. 30.
[1930]. Hyperbolic functions are extremely useful in every branch of pure physics and in the applications of physics whether to observational and experimental sciences or to technology. Thus whenever an entity (such as light, velocity, electricity, or radio-activity) is subject to gradual absorption or extinction, the decay is represented by some form of hyperbolic functions. Mercator’s projection is likewise computed by hyperbolic functions. Whenever mechanical strains are regarded great enough to be measured they are most simply expressed in terms of hyperbolic functions. Hence geological deformations invariably lead to such expressions....—Walcott, C. D.
Smithsonian Mathematical Tables, [Hyperbolic] Functions (Washington, 1909), Advertisement.
[1931]. Geometry may sometimes appear to take the lead over analysis, but in fact precedes it only as a servant goes before his master to clear the path and light him on the way. The interval between the two is as wide as between empiricism and science, as between the understanding and the reason, or as between the finite and the infinite.—Sylvester, J. J.